Hi Everybody.

Our second unit is on dynamics and chaos.

Dynamics is how things change, as I

described earlier, a fundamental aspect

of understanding complex systems is

characterizing their dynamics. How their

complex behavior unfolds and how it

changes over time. First, I will give a

brief history of the science of dynamics.

Then we will look at the notion of

iteration, and how iteration of simple

behaviors can give rise to complex

patterns. We will spend some time

discussing the crucial notion of

non-linearity and how non-linear

interactions form the crux of complex

systems. We will then look in depth at

a simple model of population growth that

gives rise to unexpected behavior. I will

explain what is meant by "chaos",

otherwise known as sensitive dependence on

initial conditions.

Dynamics is the general study of how

systems change over time. I will show

you some examples of the kinds of topics

people study under the rubrick of

dynamics. Planetary dynamics studies the

movement of planets under the force of

gravity and characterizes their orbits,

deviations from their orbits, eclipses

and so on. Fluid dynamics studies flow of

fluids and includes things like study of

ocean flows, hurricanes, gas clouds in

space and turbulent air flow like the kind

we might experience when we are flying in

an airplane. Electrical dynamics studies

the flow of electricity in circuits,

Climate dynamics looks at how climate

changes over time in terms of temperature

pressure and so on. Crowd dynamics look

at how crowds of people act. That can

either be in an ordered way or in a

disordered way, for example, when someone

calls, "Fire!" in a crowded room, people

might stampede. Population dynamics

looks at how populations vary over time.

We will be talking about that quite a bit

in this unit. Financial dynamics looks

at phenomena related to stock prices or

other financial activity.

Group dynamics looks at how groups of

animals or humans form and how they work

together to accomplish tasks. There is

also work on social dynamics. That

includes the dynamics of conflicts and of

cooperation; for example, among nations.

Dynamics is a very general field.

It has been a huge triumph of mathematics

and science to develop quantitative

tools, such as differential equations

that can be applied to explain so many

different phenomena and dynamical systems

theory is the general area of mathematics

concerned with dynamical systems, in short

it's the branch of mathematics which

describes how systems change over time

and it includes many sub-branches

including calculus, differential equations

iterated maps and so on. We'll talk

about some of this during this unit.

The dynamics of a system refers to the

manner in which the system changes.

Dynamical systems theory gives us a

vocabulary and set of mathematical tools

for describing dynamics. Let me briefly

give some history of dynamical systems

theory and mention some of the historical

big names. In the West, the study of

dynamical systems really started with

Aristotle. Aristotle believed that there

are two sets of laws. One set for earth

where objects move in straight lines,

according to Aristotle, and only under

force. Things fall to the ground at a

rate depending on how heavy they are. He

believed that there are a separate set of

physical laws for the heavens.

For example, other planets and the sun

orbit in perfect circles around the earth.

Aristotle based his views on logic and

common sense and some naive observation.

He didn't really see the need to do

systematic experiments. Only after a

couple of thousands of years did people

start questioning his views.

Nicholas Copernicus for example proposed

a new set of laws for the heavens. In his

theory the sun is stationary and the

planets orbit around it. Galileo was a

pioneer of the experimental method,

as far as studying motion was concerned.

He proved experimentally that most of

Aristotle's laws of motion were false.

Isaac Newton was the founder of the modern

science of dynamics. Newton discovered

much of what we use today to understand

the physics of motion under the force of

gravity. He proposed the then radical

view that the laws of motion are the same

on earth and in the heavens. That is,

gravity is a universal force that acts

the same no matter where in the universe.

Newton, along with Leibnitz, also invented

the branch of mathematics that we call

calculus which has been the primary tool

used ever since to study how systems

change over time and space.

Pierre-Simon Laplace was a big proponent

of Newtonian reductionism and determinism.

I will read a very famous quotation from

him which sums up his view of a

deterministic universe in which everything

is knowable in principle. Laplace said,

We may regard the present state of the

universe as the effect of its past and the

cause of its future, an intellect which at

a certain moment would know all forces

that set nature in motion and all

positions of items of which nature is

composed. If this intellect were also

vast enough to submit these data to

analysis it would embrace in a single

formula the movements of the greatest

bodies of the universe and those of the

tiniest atoms. For such an intellect

nothing would be uncertain, and the future

just like the past would be present before

its eyes. While Laplace wrote this in

the early 1800's, we could imagine today

that the intellect might be a computer,

a supercomputer that would be able to

model all of the particles in the universe

for example and the forces that act on

them and would be able to predict anything

at all. This view of the possibility of

complete prediction was widely accepted

until the late 19th or early 20th century.

Although before that Henri Poincare,

French mathematician, started to speculate

on possible reasons why such perfect

prediction might not be possible. He was

a pioneer of modern dynamical systems

theory and the notion of chaos. Let me

give his most famous quotation,

Poincare said, If we knew exactly the laws

of nature and the situation of the

universe at the initial moment, we could

predict exactly the situation of that same

universe at a succeeding moment. Now that

goes along with what Laplace said. But,

then he goes on, But even if it were the

case that the natural laws no longer had

any secret for us, we could still only

know the initial situation approximately.

If that enabled us to predict the

succeeding situation with the same

approximation, that's all we require and

we should say that the phenomenon has

been predicted. That is it governed by

laws. But it is not always so. It may

happen that small differences in the

initial conditions produce very great ones

in the final phenomena. A small error in

the former will produce an enormous error

in the latter. Prediction becomes

impossible. This quotation introduces the

notion called sensitive dependence on

initial conditions. Consider Laplace's

idea. If at a precise time, you know

the exact position and velocity of every

atom in the universe, you could use

Newton's laws to predict exactly what the

positions and velocities of the atoms will

be at some precise time in the future.

But, suppose you don't quite know the

positions exactly. Suppose you know them

only to a finite number of decimal places.

What Poincare is saying is that there are

some systems, not all, but some, in which

if you got some decimal place wrong for

position or velocity, your calculation

will eventually end up being way off.

This could be the 10th decimal place, the

hundredth, the thousandth, or however

far out you want to go. The system, if

it is sensitive to initial conditions,

where the initial conditions, for

example, would be the positions and

velocities of every atom, at some

particular time, if it is sensitive in

that way, then if you don't know exact

values, for the initial conditions,

prediction becomes impossible. Poincare's

notion of sensitive dependence on initial

conditions, is illustrated by the famous

so called butterfly effect. In this

hypothetical example, a small butterfly

flaps its wings in Tokyo. This causes a

change in the position and velocity of a

few air molecules. If the whole weather

system is sensitive to initial conditions

and the weather forecasters don't take the

butterfly into account, after some time,

their predictions will be way off, and the

butterfly might in fact create a hurricane

This is not to say that this actually

happens, or the weather is sensitive

to initial conditions, Poincare is just

saying that there are some systems with

this property, and we don't know what

they all are. We will look at a simple

one a bit later in this unit. Now we can

define the notion of chaos. Chaos is used

in colloquial every day language to mean

roughly disorder, but in dynamical

systems theory, it means something

specific. It is one particular type of

the dynamics of a system. It is one way

in which systems change. It is defined

as sensitive dependence on initial

conditions. We will make this quite a

bit more precise later on. Now you might

be familiar with this fellow,

Dr. Ian Malcolm, who asked, "You've never

heard of chaos theory? Non-linear

equations? Strange attractors?"

If this is sounding familiar, you might

have actually seen him. He was a

character in a book and then a movie,

called Jurassic Park, back in the 90's.

You may or may not know, that the sequel

to Jurassic Park, also written by Michael

Creighton, and made into a movie was

called The Lost World.

Part of The Lost World took place at the

Santa Fe Institute. In the Prologue, we

have "Life at the Edge of Chaos",

Creighton writes that the Santa Fe

Institute was housed in a series of

buildings on Canyon Road which had

formerly been a convent. That is actually

true. The Institute seminars were held

in a room which had served as a chapel

which is also true. Now standing at the

podium with a shaft of sunlight shining

down on him, Ian Malcolm, paused

dramatically before continuing his lecture

dressed entirely in black, leaning on a

cane, Malcolm gave the impression of

severity. He was known in the Institute

for his unconventional analysis and his

tendency to pessimism. His talk that

August entitled, Life at the Edge of Chaos

was typical of his thinking. In it,

Malcolm presented his analysis of chaos

theory as it applied to evolution.

Once this book came out, and the movie

came out, a lot of people noticed that

there was a place called the Santa Fe

Institute. I happened to be there in the

mid-90's, as a resident faculty member,

and one day the Santa Fe Institute

librarian came to lunch and was sitting

with a group of faculty and post-docs at

the Institute and mentioned humorously

that someone had written her a letter

requesting Professor Ian Malcolm's

papers. So, of course, post-docs, being

post-docs, decided to do the obvious thing

which was make Ian Malcolm a web site.

Here is his web site at the Santa Fe

Institute, which had some of his papers

in it, and his research interests and so

on, and only after the Board of Trustees

of the Santa Fe Institute decided that

this was unprofessional did Ian Malcolm's

web page come down. Chaos is a very

important area in dynamical systems

theory and shows up in many different

contexts. You can see all these different

areas in which chaos has been seen such as

brain activity, population growth,

financial data, and so on.

We're going to look at the phenomenon of

chaos in population growth, in a very

simple model of population growth.

We're going to address the question,

What is the difference between chaos and

randomness? Which turns out to be a more

subtle question than one might think.

We're going to explore it through the

notion of "deterministic chaos".